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Table of Contents
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Table of Contents
 * What Is a Monte Carlo Simulation?
 * How Does the Monte Carlo Simulation Assess Risk?
 * What Is the History of the Monte Carlo Simulation?
 * How Does the Monte Carlo Simulation Method Work?
 * The 4 Steps in a Monte Carlo Simulation
 * Monte Carlo Simulation Results Explained
 * Advantages and Disadvantages of a Monte Carlo Simulation
 * Monte Carlo Simulation FAQs
 * The Bottom Line

 * Corporate Finance
 * Financial Analysis


MONTE CARLO SIMULATION: HISTORY, HOW IT WORKS, AND 4 KEY STEPS




By
Will Kenton

Full Bio
 * 

Will Kenton is an expert on the economy and investing laws and regulations. He
previously held senior editorial roles at Investopedia and Kapitall Wire and
holds a MA in Economics from The New School for Social Research and Doctor of
Philosophy in English literature from NYU.
Learn about our editorial policies
Updated November 02, 2023
Reviewed by Margaret James
Fact checked by
Suzanne Kvilhaug
Fact checked by Suzanne Kvilhaug
Full Bio
Suzanne is a content marketer, writer, and fact-checker. She holds a Bachelor of
Science in Finance degree from Bridgewater State University and helps develop
content strategies for financial brands.
Learn about our editorial policies


WHAT IS A MONTE CARLO SIMULATION?

A Monte Carlo simulation is used to model the probability of different outcomes
in a process that cannot easily be predicted due to the intervention of random
variables. It is a technique used to understand the impact of risk and
uncertainty.



A Monte Carlo simulation is used to tackle a range of problems in many fields
including investing, business, physics, and engineering. It is also referred to
as a multiple probability simulation.




KEY TAKEAWAYS

 * A Monte Carlo simulation is a model used to predict the probability of a
   variety of outcomes when the potential for random variables is present.
 * Monte Carlo simulations help to explain the impact of risk and uncertainty in
   prediction and forecasting models.
 * A Monte Carlo simulation requires assigning multiple values to an uncertain
   variable to achieve multiple results and then averaging the results to obtain
   an estimate.
 * Monte Carlo simulations assume perfectly efficient markets.



Investopedia / Eliana Rodgers


HOW DOES THE MONTE CARLO SIMULATION ASSESS RISK?

When faced with significant uncertainty in making a forecast or estimate, some
methods replace the uncertain variable with a single average number. The Monte
Carlo Simulation instead uses multiple values and then averages the results.



Monte Carlo simulations have a vast array of applications in fields that are
plagued by random variables, notably business and investing. They are used to
estimate the probability of cost overruns in large projects and the likelihood
that an asset price will move in a certain way.



Telecoms use them to assess network performance in various scenarios, which
helps them to optimize their networks. Financial analysts use Monte Carlo
simulations to assess the risk that an entity will default, and to analyze
derivatives such as options. Insurers and oil well drillers also use them to
measure risk.

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Monte Carlo simulations have many applications outside of business and finance,
such as in meteorology, astronomy, and particle physics.




WHAT IS THE HISTORY OF THE MONTE CARLO SIMULATION?

The Monte Carlo simulation was named after the gambling destination in Monaco
because chance and random outcomes are central to this modeling technique, as
they are to games like roulette, dice, and slot machines.



The technique was initially developed by Stanislaw Ulam, a mathematician who
worked on the Manhattan Project, the secret effort to create the first atomic
weapon. He shared his idea with John Von Neumann, a colleague at the Manhattan
Project, and the two collaborated to refine the Monte Carlo simulation.1




HOW DOES THE MONTE CARLO SIMULATION METHOD WORK?

The Monte Carlo method acknowledges an issue for any simulation technique: the
probability of varying outcomes cannot be firmly pinpointed because of random
variable interference. Therefore, a Monte Carlo simulation focuses on constantly
repeating random samples.



A Monte Carlo simulation takes the variable that has uncertainty and assigns it
a random value. The model is then run and a result is provided. This process is
repeated again and again while assigning many different values to the variable
in question. Once the simulation is complete, the results are averaged to arrive
at an estimate.




THE 4 STEPS IN A MONTE CARLO SIMULATION

To perform a Monte Carlo simulation, there are four main steps. Microsoft Excel
or a similar program can be used to create a Monte Carlo simulation that
estimates the probable price movements of stocks or other assets.



There are two components to an asset's price movement: drift, which is its
constant directional movement, and a random input, which represents market
volatility.



By analyzing historical price data, you can determine the drift, standard
deviation, variance, and average price movement of a security. These are the
building blocks of a Monte Carlo simulation.



To create a Monte Carlo simulation, consider the following four steps:



Step 1: To project one possible price trajectory, use the historical price data
of the asset to generate a series of periodic daily returns using the natural
logarithm (note that this equation differs from the usual percentage change
formula):



> Periodic Daily Return=ln(Day’s PricePrevious Day’s Price)\begin{aligned}
> &\text{Periodic Daily Return} = ln \left ( \frac{ \text{Day's Price} }{
> \text{Previous Day's Price} } \right ) \\ \end{aligned}
> Periodic Daily Return=ln(Previous Day’s PriceDay’s Price )



Step 2: Next use the AVERAGE, STDEV.P, and VAR.P functions on the entire
resulting series to obtain the average daily return, standard deviation, and
variance inputs, respectively. The drift is equal to:



> Drift=Average Daily Return−Variance2where:Average Daily Return=Produced from Excel’sAVERAGE function from periodic daily returns seriesVariance=Produced from Excel’sVAR.P function from periodic daily returns series\begin{aligned}
> &\text{Drift} = \text{Average Daily Return} - \frac{ \text{Variance} }{ 2 } \\
> &\textbf{where:} \\ &\text{Average Daily Return} = \text{Produced from
> Excel's} \\ &\text{AVERAGE function from periodic daily returns series} \\
> &\text{Variance} = \text{Produced from Excel's} \\ &\text{VAR.P function from
> periodic daily returns series} \\ \end{aligned}
> Drift=Average Daily Return−2Variance
> where:Average Daily Return=Produced from Excel’sAVERAGE function from periodic daily returns seriesVariance=Produced from Excel’sVAR.P function from periodic daily returns series



Alternatively, drift can be set to 0; this choice reflects a certain theoretical
orientation, but the difference will not be huge, at least for shorter time
frames.



Step 3: Next, obtain a random input:



> Random Value=σ×NORMSINV(RAND())where:σ=Standard deviation, produced from Excel’sSTDEV.P function from periodic daily returns seriesNORMSINV and RAND=Excel functions\begin{aligned}
> &\text{Random Value} = \sigma \times \text{NORMSINV(RAND())} \\
> &\textbf{where:} \\ &\sigma = \text{Standard deviation, produced from Excel's}
> \\ &\text{STDEV.P function from periodic daily returns series} \\
> &\text{NORMSINV and RAND} = \text{Excel functions} \\ \end{aligned}
> Random Value=σ×NORMSINV(RAND())where:σ=Standard deviation, produced from Excel’sSTDEV.P function from periodic daily returns seriesNORMSINV and RAND=Excel functions



The equation for the following day's price is:



> Next Day’s Price=Today’s Price×e(Drift+Random Value)\begin{aligned}
> &\text{Next Day's Price} = \text{Today's Price} \times e^{ ( \text{Drift} +
> \text{Random Value} ) }\\ \end{aligned}
> Next Day’s Price=Today’s Price×e(Drift+Random Value)




Step 4: To take e to a given power x in Excel, use the EXP function:
EXP(x). Repeat this calculation the desired number of times. (Each repetition
represents one day.) The result is a simulation of the asset's future price
movement.



By generating an arbitrary number of simulations, you can assess the probability
that a security's price will follow a given trajectory.




MONTE CARLO SIMULATION RESULTS EXPLAINED

The frequencies of different outcomes generated by this simulation will form a
normal distribution, that is, a bell curve. The most likely return is in the
middle of the curve, meaning there is an equal chance that the actual return
will be higher or lower.



The probability that the actual return will be within one standard deviation of
the most probable ("expected") rate is 68%. The probability that it will be
within two standard deviations is 95%, and that it will be within three standard
deviations 99.7%. 



Still, there is no guarantee that the most expected outcome will occur, or
that actual movements will not exceed the wildest projections.



Crucially, a Monte Carlo simulation ignores everything that is not built into
the price movement such as macro trends, a company's leadership, market hype,
and cyclical factors).



In other words, it assumes a perfectly efficient market.




ADVANTAGES AND DISADVANTAGES OF A MONTE CARLO SIMULATION

The Monte Carlo method is used to help an investor estimate the likelihood of a
gain or a loss on a certain investment. Other methods have the same aim.



The Monte Carlo simulation was created to overcome a perceived disadvantage of
other methods of estimating a probable outcome.



No simulation can pinpoint an inevitable outcome. The Monte Carlo method aims at
a sounder estimate of the probability that an outcome will differ from a
projection.



The difference is that the Monte Carlo method tests a number of random variables
and then averages them, rather than starting out with an average.



Like any financial simulation, the Monte Carlo method uses historical price data
as the basis for a projection of future price data. It then disrupts the pattern
by introducing random variables, represented by numbers. Finally, it averages
those numbers to arrive at an estimate of the risk that the pattern will be
disrupted in real life.




HOW IS THE MONTE CARLO SIMULATION USED IN FINANCE APPLICATIONS?

The Monte Carlo simulation is used to estimate the probability of a certain
income. As such, it is widely used by investors and financial analysts to
evaluate the probable success of investments they're considering. Some common
uses include:

 * Pricing stock options. The potential price movements of the underlying asset
   are tracked given every possible variable. The results are averaged and then
   discounted to the asset's current price. This is intended to indicate the
   probable payoff of the options.2
 * Portfolio valuation. A number of alternative portfolios can be tested using
   the Monte Carlo simulation in order to arrive at a measure of their
   comparative risk.2
 * Fixed income investments. The short rate is the random variable here. The
   simulation is used to calculate the probable impact of movements in the short
   rate on fixed rate investments.2
   




WHAT PROFESSIONS USE THE MONTE CARLO SIMULATION?

It may be best known for its financial applications, but the Monte Carlo
simulation is used in virtually every profession that must measure risks and
prepare to meet them.

For example, a telecom may build its network to sustain all of its users all of
the time. In order to do that, it must consider all of the possible variations
in demand for the service. It must determine whether the system will stand the
strain of peak hours and peak seasons.

A Monte Carlo simulation may help the telecom company decide whether its service
is likely to stand the strain of Super Bowl Sunday as well as an average Sunday
in August.




WHAT FACTORS ARE EVALUATED IN A MONTE CARLO SIMULATION?

A Monte Carlo simulation in investing is based on historical price data on the
asset or assets being evaluated.

The building blocks of the simulation, derived from the historical data, are
drift, standard deviation, variance, and average price movement.




THE BOTTOM LINE

The Monte Carlo simulation shows the spectrum of probable outcomes for an
uncertain scenario. This technique assigns multiple values to uncertain
variables, obtains multiple results, and then takes the average of these results
to arrive at an estimate.



From investing to engineering, the Monte Carlo method is used in many
applications to measure risk including estimating the likelihood of a gain or
loss in an investment, or the odds that a project will run over budget.




Article Sources

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in producing accurate, unbiased content in our editorial policy.

 1. Virginia Polytechnic Institute. "Monte Carlo Simulation."

 2. Corporate Finance Institute. "Monte Carlo Simulation"



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